Multi-objective equilibrium optimizer slime mould algorithm and its application in solving engineering problems

Abstract

This paper aims to represent a multi-objective equilibrium optimizer slime mould algorithm (MOEOSMA) to solve real-world constraint engineering problems. The proposed algorithm has a better optimization performance than the existing multi-objective slime mould algorithm. In the MOEOSMA, dynamic coefficients are used to adjust exploration and exploitation trends. The elite archiving mechanism is used to promote the convergence of the algorithm. The crowding distance method is used to maintain the distribution of the Pareto front. The equilibrium pool strategy is used to simulate the cooperative foraging behavior of the slime mould, which helps to enhance the exploration ability of the algorithm. The performance of MOEOSMA is evaluated on the latest CEC2020 functions, eight real-world multi-objective constraint engineering problems, and four large-scale truss structure optimization problems. The experimental results show that the proposed MOEOSMA not only finds more Pareto optimal solutions, but also maintains a good distribution in the decision space and objective space. Statistical results show that MOEOSMA has a strong competitive advantage in terms of convergence, diversity, uniformity, and extensiveness, and its comprehensive performance is significantly better than other comparable algorithms.

Appendices

Appendix 1. Real-world constraint engineering design problems

1.1 Speed reducer design problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} ] = [b,m,z,l_{1} ,l_{2} ,d_{1} ,d_{2} ] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = 0.7854x_{1} x_{2}^{2} (14.9334x_{3} + 3.3333x_{3}^{2} - 43.0934) \hfill \\ \, - 1.508x_{1} \left( {x_{6}^{2} + x_{7}^{2} } \right) + 0.7854\left( {x_{4} x_{6}^{2} + x_{5} x_{7}^{2} } \right) + 7.4777\left( {x_{6}^{3} + x_{7}^{3} } \right) \hfill \\ \, f_{2} ({\mathbf{x}}) = {{\sqrt {\left( {{{745x_{4} } \mathord{\left/ {\vphantom {{745x_{4} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 16.9 \times 10^{6} } } \mathord{\left/ {\vphantom {{\sqrt {\left( {{{745x_{4} } \mathord{\left/ {\vphantom {{745x_{4} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 16.9 \times 10^{6} } } {(0.1x_{6}^{3} )}}} \right. \kern-0pt} {(0.1x_{6}^{3} )}} \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = {{27} \mathord{\left/ {\vphantom {{27} {(x_{1} x_{3} x_{2}^{2} )}}} \right. \kern-0pt} {(x_{1} x_{3} x_{2}^{2} )}} \le 1; \, g_{2} ({\mathbf{x}}) = {{397.5} \mathord{\left/ {\vphantom {{397.5} {(x_{1} x_{2}^{2} x_{3}^{2} )}}} \right. \kern-0pt} {(x_{1} x_{2}^{2} x_{3}^{2} )}} \le 1 \hfill \\ \, g_{3} ({\mathbf{x}}) = {{1.93x_{4}^{3} } \mathord{\left/ {\vphantom {{1.93x_{4}^{3} } {(x_{2} x_{3} x_{6}^{4} )}}} \right. \kern-0pt} {(x_{2} x_{3} x_{6}^{4} )}} \le 1; \, g_{4} ({\mathbf{x}}) = {{1.93x_{5}^{3} } \mathord{\left/ {\vphantom {{1.93x_{5}^{3} } {(x_{2} x_{3} x_{7}^{4} )}}} \right. \kern-0pt} {(x_{2} x_{3} x_{7}^{4} )}} \le 1 \hfill \\ \, g_{5} ({\mathbf{x}}) = {{x_{2} x_{3} } \mathord{\left/ {\vphantom {{x_{2} x_{3} } {40}}} \right. \kern-0pt} {40}} \le 1; \, g_{6} ({\mathbf{x}}) = {{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {(12x_{2} )}}} \right. \kern-0pt} {(12x_{2} )}} \le 1 \hfill \\ \, g_{7} ({\mathbf{x}}) = {{5x_{2} } \mathord{\left/ {\vphantom {{5x_{2} } {x_{1} }}} \right. \kern-0pt} {x_{1} }} \le 1; \, g_{8} ({\mathbf{x}}) = {{(1.5x_{6} + 1.9)} \mathord{\left/ {\vphantom {{(1.5x_{6} + 1.9)} {x_{4} }}} \right. \kern-0pt} {x_{4} }} \le 1 \hfill \\ \, g_{9} ({\mathbf{x}}) = {{(1.1x_{7} + 1.9)} \mathord{\left/ {\vphantom {{(1.1x_{7} + 1.9)} {x_{5} }}} \right. \kern-0pt} {x_{5} }} \le 1; \, g_{10} ({\mathbf{x}}) = f_{2} ({\mathbf{x}}) - 1100 \le 1 \hfill \\ \, g_{11} ({\mathbf{x}}) = {{\sqrt {\left( {{{745x_{5} } \mathord{\left/ {\vphantom {{745x_{5} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 157.5 \times 10^{6} } } \mathord{\left/ {\vphantom {{\sqrt {\left( {{{745x_{5} } \mathord{\left/ {\vphantom {{745x_{5} } {(x_{2} x_{3} )}}} \right. \kern-0pt} {(x_{2} x_{3} )}}} \right)^{2} + 157.5 \times 10^{6} } } {(0.1x_{7}^{3} )}}} \right. \kern-0pt} {(0.1x_{7}^{3} )}} - 850 \le 1 \hfill \\ {\text{with 2}}{.6} \le x_{1} \le 3.6,0.7 \le x_{2} \le 0.8,17 \le x_{3} \le 28({\text{integer}}), \hfill \\ \, 7.3 \le x_{4} ,x_{5} \le 8.3,{2}{\text{.9}} \le x_{6} \le 3.9,5.0 \le x_{7} \le 5.5. \hfill \\ \end{gathered}$$

1.2 Spring design problem

$$\begin{gathered} {\text{Consider: }}{\mathbf{x}} = [x_{1} ,x_{2} ,x_{3} ] = [d,D,N] \hfill \\ {\text{Minimize: }}f_{1} ({\mathbf{x}}) = 0.25\pi^{2} x_{1}^{2} x_{2} (x_{3} + 2) \hfill \\ \, f_{2} ({\mathbf{x}}) = 8000{{c_{f} x_{2} } \mathord{\left/ {\vphantom {{c_{f} x_{2} } {(\pi x_{1}^{3} )}}} \right. \kern-0pt} {(\pi x_{1}^{3} )}} \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = f_{1} ({\mathbf{x}}) - V_{\max } \le 0; \, g_{2} ({\mathbf{x}}) = f_{2} ({\mathbf{x}}) - S \le 0 \hfill \\ \, g_{3} ({\mathbf{x}}) = l_{f} - l_{\max } \le 0; \, g_{4} ({\mathbf{x}}) = d_{\min } - x_{1} \le 0 \hfill \\ \, g_{5} ({\mathbf{x}}) = x_{1} + x_{2} - D_{\max } \le 0; \, g_{6} ({\mathbf{x}}) = 3 - C \le 0 \hfill \\ \, g_{7} ({\mathbf{x}}) = \sigma_{p} - \sigma_{pm} \le 0; \, g_{8} ({\mathbf{x}}) = 1.25 - {{700} \mathord{\left/ {\vphantom {{700} K}} \right. \kern-0pt} K} \le 0 \hfill \\ {\text{where }}V_{\max } = 30; \, S = 189000; \, l_{\max } = 14; \, d_{\min } = 0.2; \hfill \\ \, D_{\max } = 3; \, \sigma_{pm} = 6; \, \sigma_{p} = {{300} \mathord{\left/ {\vphantom {{300} K}} \right. \kern-0pt} K}; \hfill \\ \, K = {{Gx_{1}^{4} } \mathord{\left/ {\vphantom {{Gx_{1}^{4} } {(8x_{2}^{3} x_{3} )}}} \right. \kern-0pt} {(8x_{2}^{3} x_{3} )}}; \, G = 11.5 \times 10^{6} ; \hfill \\ \, c_{f} = {{(4C - 1)} \mathord{\left/ {\vphantom {{(4C - 1)} {(4C - 4)}}} \right. \kern-0pt} {(4C - 4)}} + {{0.615} \mathord{\left/ {\vphantom {{0.615} C}} \right. \kern-0pt} C}; \, C = {{x_{2} } \mathord{\left/ {\vphantom {{x_{2} } {x_{1} }}} \right. \kern-0pt} {x_{1} }}; \hfill \\ \, l_{f} = 1.05x_{1} (x_{3} + 2) + {{1000} \mathord{\left/ {\vphantom {{1000} K}} \right. \kern-0pt} K}. \hfill \\ {\text{with }}x_{1} \in \{ {0}{\text{.009, 0}}{.0095, 0}{\text{.0104, 0}}{.0118, 0}{\text{.0128, 0}}{.0132, 0}{\text{.014,}} \hfill \\ { 0}{\text{.015, 0}}{.0162, 0}{\text{.0173, 0}}{.018, 0}{\text{.020, 0}}{.023, 0}{\text{.025,}} \hfill \\ { 0}{\text{.028, 0}}{.032, 0}{\text{.035, 0}}{.041, 0}{\text{.047, 0}}{.054, 0}{\text{.063,}} \hfill \\ { 0}{\text{.072, 0}}{.080, 0}{\text{.092, 0}}{.0105, 0}{\text{.120, 0}}{.135, 0}{\text{.148,}} \hfill \\ { 0}{\text{.162, 0}}{.177, 0}{\text{.192, 0}}{.207, 0}{\text{.225, 0}}{.244, 0}{\text{.263,}} \hfill \\ { 0}{\text{.283, 0}}{.307, 0}{\text{.331, 0}}{.362, 0}{\text{.394, 0}}{.4375, 0}{\text{.500}}\} , \hfill \\ { 1} \le x_{2} \le 30({\text{continuous}}),1 \le x_{3} \le 32({\text{integer}}). \hfill \\ \end{gathered}$$

1.3 Hydrostatic thrust bearing design problem

$$\begin{gathered} {\text{Consider: }}{\mathbf{x}} = [R,R_{0} ,\mu ,Q] \hfill \\ {\text{Minimize: }}f_{1} ({\mathbf{x}}) = \frac{1}{12}\left( {\frac{{Q \times P_{0} }}{0.7} + E_{f} } \right) \hfill \\ \, f_{2} ({\mathbf{x}}) = \frac{\gamma }{{g \cdot P_{0} }} \cdot \frac{Q}{2\pi Rh} \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = W_{s} - W \le 0; \, g_{2} ({\mathbf{x}}) = P_{0} - P_{\max } \le 0 \hfill \\ \, g_{3} ({\mathbf{x}}) = \Delta T - \Delta T_{\max } \le 0; \, g_{4} ({\mathbf{x}}) = h_{\min } - h \le 0 \hfill \\ \, g_{5} ({\mathbf{x}}) = R_{0} - R \le 0; \, g_{6} ({\mathbf{x}}) = f_{2} ({\mathbf{x}}) - 0.001 \le 0 \hfill \\ \, g_{7} ({\mathbf{x}}) = {W \mathord{\left/ {\vphantom {W {\left( {\pi \left( {R^{2} - R_{0}^{2} } \right)} \right)}}} \right. \kern-0pt} {\left( {\pi \left( {R^{2} - R_{0}^{2} } \right)} \right)}} - 5000 \le 0 \hfill \\ {\text{where }}W = \frac{{\pi P_{0} }}{2} \cdot \frac{{R^{2} - R_{0}^{2} }}{{\ln \left( {{R \mathord{\left/ {\vphantom {R {R_{0} }}} \right. \kern-0pt} {R_{0} }}} \right)}}; \, P_{0} = \frac{6\mu Q}{{\pi h^{3} }} \cdot \ln \left( {\frac{R}{{R_{0} }}} \right); \hfill \\ \, h = \left( {\frac{2\pi N}{{60}}} \right)^{2} \cdot \frac{2\pi \mu }{{E_{f} }} \cdot \frac{{R^{4} - R_{0}^{4} }}{4}; \, E_{f} = 9336Q \cdot \gamma \cdot C \cdot \Delta T; \hfill \\ \, \Delta T = 2\left( {10^{P} - 560} \right); \, P = \frac{{\log_{10} \log_{10} \left( {8.122 \times 10^{6} \mu + 0.8} \right) - C_{1} }}{n}; \hfill \\ \, \gamma = 0.0307; \, C = 0.5; \, n = - 3.55; \, C_{1} = 10.04; \, W_{s} = 101000; \hfill \\ \, P_{\max } = 1000; \, \Delta T_{\max } = 50; \, h_{\min } = 0.001; \, g = 386.4; \, N = 750. \hfill \\ {\text{with 1}} \le R,R_{0} ,Q \le 16,1 \times {10}^{ - 6} \le \mu \le 16 \times {10}^{ - 6} . \hfill \\ \end{gathered}$$

1.4 Vibrating platform design problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [d_{1} ,d_{2} ,d_{3} ,b,L] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = - {\pi \mathord{\left/ {\vphantom {\pi {(2L^{2} )}}} \right. \kern-0pt} {(2L^{2} )}} \cdot \sqrt {{{EI} \mathord{\left/ {\vphantom {{EI} \mu }} \right. \kern-0pt} \mu }} \hfill \\ \, f_{2} ({\mathbf{x}}) = 2b \cdot L\left( {c_{1} d_{1} - c_{2} (d_{1} - d_{2} ) - c_{3} (d_{2} - d_{3} )} \right) \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = \mu L - 2800 \le 0; \, g_{2} ({\mathbf{x}}) = d_{1} - d_{2} \le 0 \hfill \\ \, g_{3} ({\mathbf{x}}) = d_{2} - d_{1} - 0.15 \le 0; \, g_{4} ({\mathbf{x}}) = d_{2} - d_{3} \le 0 \hfill \\ \, g_{5} ({\mathbf{x}}) = d_{3} - d_{2} - 0.01 \le 0 \hfill \\ {\text{where }}EI = ({{2b} \mathord{\left/ {\vphantom {{2b} 3}} \right. \kern-0pt} 3})(E_{1} d_{1}^{3} - E_{2} (d_{1}^{3} - d_{2}^{3} ) - E_{3} (d_{2}^{3} - d_{3}^{3} )); \hfill \\ \, \mu = 2b(\rho_{1} d_{1} - \rho_{2} (d_{1} - d_{2} ) - \rho_{3} (d_{2} - d_{3} )). \hfill \\ {\text{with 0}}{.05} \le d_{1} \le 0.5,0.2 \le d_{2} \le 0.5,0.2 \le d_{3} \le 0.6, \hfill \\ \, 0.35 \le b \le 0.5,{3} \le L \le 6. \hfill \\ \end{gathered}$$

1.5 Car side impact design problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} ] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = 4.90x_{1} + 6.67x_{2} + 6.98x_{3} + 4.01x_{4} + 1.78x_{5} \hfill \\ \, + 10^{ - 5} x_{6} + 2.73x_{7} + 1.98 \hfill \\ \, f_{2} ({\mathbf{x}}) = 4.72 - 0.19x_{2} x_{3} - 0.50x_{4} \hfill \\ \, f_{3} ({\mathbf{x}}) = 0.50 \cdot (V_{MBP} + V_{FD} ) \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = 1.16 - 0.0092928x_{3} - 0.3717x_{2} x_{4} \le 1 \hfill \\ \, g_{2} ({\mathbf{x}}) = 0.261 - 0.06486x_{1} + 0.0154464x_{6} - 0.0159x_{1} x_{2} \hfill \\ \, - 0.019x_{2} x_{7} + 0.0144x_{3} x_{5} \le 0.32 \hfill \\ \, g_{3} ({\mathbf{x}}) = 0.214 - 0.0587118x_{1} + 0.018x_{2}^{2} + 0.030408x_{3} \hfill \\ \, + 0.00817x_{5} + 0.03099x_{2} x_{6} - 0.018x_{2} x_{7} \hfill \\ \, - 0.00364x_{5} x_{6} \le 0.32 \hfill \\ \, g_{4} ({\mathbf{x}}) = 0.74 - 0.61x_{2} + 0.227x_{2}^{2} - 0.031296x_{3} \hfill \\ \, - 0.031872x_{7} \le 0.32 \hfill \\ \, g_{5} ({\mathbf{x}}) = 28.98 + 3.818x_{3} + 1.27296x_{6} - 2.68065x_{7} \hfill \\ \, - 4.2x_{1} x_{2} \le 32 \hfill \\ \, g_{6} ({\mathbf{x}}) = 33.86 - 3.795x_{2} + 2.95x_{3} - 3.4431x_{7} - 5.057x_{1} x_{2} \hfill \\ \, + 1.45728 \le 32 \hfill \\ \, g_{7} ({\mathbf{x}}) = 46.36 - 4.4505x_{1} - 9.9x_{2} \le 32 \hfill \\ \, g_{8} ({\mathbf{x}}) = f_{2} ({\mathbf{x}}) \le 4 \hfill \\ \, g_{9} ({\mathbf{x}}) = V_{MBP} \le 9.9 \hfill \\ \, g_{10} ({\mathbf{x}}) = V_{FD} \le 15.7 \hfill \\ {\text{where }}V_{MBP} = 10.58 - 0.67275x_{2} - 0.674x_{1} x_{2} ; \hfill \\ \, V_{FD} = 16.45 - 0.489x_{3} x_{7} - 0.843x_{5} x_{6} . \hfill \\ {\text{with 0}}{.5} \le x_{1} ,x_{3} ,x_{4} \le 1.5,{0}{\text{.45}} \le x_{2} \le 1.35, \hfill \\ { 0}{\text{.875}} \le x_{5} \le 2.625,0.4 \le x_{6} ,x_{7} \le 1.2. \hfill \\ \end{gathered}$$

1.6 Water resource management problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [x_{1} ,x_{2} ,x_{3} ] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = 106780.37(x_{2} + x_{3} ) + 61704.67 \hfill \\ \, f_{2} ({\mathbf{x}}) = 3000x_{1} \hfill \\ \, f_{3} ({\mathbf{x}}) = {{30570 \times 2289x_{2} } \mathord{\left/ {\vphantom {{30570 \times 2289x_{2} } {(0.06 \times 2289)^{0.65} }}} \right. \kern-0pt} {(0.06 \times 2289)^{0.65} }} \hfill \\ \, f_{4} ({\mathbf{x}}) = 250 \times 2289 \times \exp \left( {2.74 - 39.75x_{2} + 9.9x_{3} } \right) \hfill \\ \, f_{5} ({\mathbf{x}}) = 25\left( {{{1.39} \mathord{\left/ {\vphantom {{1.39} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} + 4940x_{3} - 80} \right) \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = 4.94x_{3} + {{0.00139} \mathord{\left/ {\vphantom {{0.00139} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 1.08 \hfill \\ \, g_{2} ({\mathbf{x}}) = 1.082x_{3} + {{0.000306} \mathord{\left/ {\vphantom {{0.000306} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 1.0986 \hfill \\ \, g_{3} ({\mathbf{x}}) = 49408.24x_{3} + {{12.307} \mathord{\left/ {\vphantom {{12.307} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 54051.02 \hfill \\ \, g_{4} ({\mathbf{x}}) = 8046.33x_{3} + {{2.098} \mathord{\left/ {\vphantom {{2.098} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 16696.71 \hfill \\ \, g_{5} ({\mathbf{x}}) = 7883.39x_{3} + {{2.138} \mathord{\left/ {\vphantom {{2.138} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 10705.04 \hfill \\ \, g_{6} ({\mathbf{x}}) = 1721.26x_{3} + 0.417x_{1} x_{2} \le 2136.54 \hfill \\ \, g_{7} ({\mathbf{x}}) = 631.13x_{3} + {{0.164} \mathord{\left/ {\vphantom {{0.164} {(x_{1} x_{2} )}}} \right. \kern-0pt} {(x_{1} x_{2} )}} \le 604.48 \hfill \\ {\text{with 0}}{.01} \le x_{1} \le 0.45,{0}{\text{.01}} \le x_{2} ,x_{3} \le 0.1. \hfill \\ \end{gathered}$$

1.7 Bulk carriers design problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [L,B,D,T,V_{k} ,C_{B} ] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = {{(C_{c} + C_{r} + C_{v} )} \mathord{\left/ {\vphantom {{(C_{c} + C_{r} + C_{v} )} {C_{a} }}} \right. \kern-0pt} {C_{a} }} \hfill \\ \, f_{2} ({\mathbf{x}}) = W_{ls} \hfill \\ \, f_{3} ({\mathbf{x}}) = - C_{a} \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = - {L \mathord{\left/ {\vphantom {L B}} \right. \kern-0pt} B} + 6 \le 0 \hfill \\ \, g_{2} ({\mathbf{x}}) = {L \mathord{\left/ {\vphantom {L D}} \right. \kern-0pt} D} - 15 \le 0 \hfill \\ \, g_{3} ({\mathbf{x}}) = - {L \mathord{\left/ {\vphantom {L T}} \right. \kern-0pt} T} - 19 \le 0 \hfill \\ \, g_{4} ({\mathbf{x}}) = T - 0.45D_{wt}^{0.31} \le 0 \hfill \\ \, g_{5} ({\mathbf{x}}) = T - 0.7D - 0.7 \le 0 \hfill \\ \, g_{6} ({\mathbf{x}}) = F_{n} - 0.32 \le 0 \hfill \\ \, g_{7} ({\mathbf{x}}) = - 0.53T - {{((0.085C_{B} - 0.002)B^{2} )} \mathord{\left/ {\vphantom {{((0.085C_{B} - 0.002)B^{2} )} {(T \cdot C_{B} )}}} \right. \kern-0pt} {(T \cdot C_{B} )}} \hfill \\ \, + (1 + 0.52D) + 0.07B \le 0 \hfill \\ \, g_{8} ({\mathbf{x}}) = - D_{wt} + 3000 \le 0; \hfill \\ \, g_{9} ({\mathbf{x}}) = D_{wt} - 500000 \le 0 \hfill \\ {\text{where }}C_{c} = 2.6(2000W_{s}^{0.85} + 3500W_{o} + 2400P^{0.8} ); \hfill \\ \, C_{r} = 40000D_{wt}^{0.3} ; \, C_{v} = (105D_{c} S_{d} + 6.3D_{wt}^{0.8} )R_{tpa} ; \hfill \\ \, C_{a} = D_{cwt} R_{tpa} ; \, R_{tpa} = {{350} \mathord{\left/ {\vphantom {{350} {\left( {S_{d} + 2({{D_{cwt} } \mathord{\left/ {\vphantom {{D_{cwt} } {8000}}} \right. \kern-0pt} {8000}} + 0.5)} \right)}}} \right. \kern-0pt} {\left( {S_{d} + 2({{D_{cwt} } \mathord{\left/ {\vphantom {{D_{cwt} } {8000}}} \right. \kern-0pt} {8000}} + 0.5)} \right)}}; \hfill \\ \, D_{cwt} = D_{wt} - D_{c} (S_{d} + 5) - 2D_{wt}^{0.5} ; \, S_{d} = {{5000V_{k} } \mathord{\left/ {\vphantom {{5000V_{k} } {24}}} \right. \kern-0pt} {24}}; \hfill \\ \, D_{c} = {{0.19 \times 24P} \mathord{\left/ {\vphantom {{0.19 \times 24P} {1000}}} \right. \kern-0pt} {1000}} + 0.2; \, D_{wt} = 1.025L \cdot B \cdot T \cdot C_{B} - W_{ls} ; \hfill \\ \, W_{ls} = W_{s} + W_{o} + W_{m} ; \, W_{s} = 0.034L^{1.7} B^{0.7} D^{0.4} C_{B}^{0.5} ; \hfill \\ \, W_{o} = L^{0.8} B^{0.6} D^{0.3} C_{B}^{0.1} ; \, W_{m} = 0.17P^{0.9} ; \hfill \\ \, P = {{(1.025L \cdot B \cdot T \cdot C_{B} )^{{\tfrac{2}{3}}} V_{k}^{3} } \mathord{\left/ {\vphantom {{(1.025L \cdot B \cdot T \cdot C_{B} )^{{\tfrac{2}{3}}} V_{k}^{3} } {(a + b \cdot F_{n} )}}} \right. \kern-0pt} {(a + b \cdot F_{n} )}}; \hfill \\ \, F_{n} = {{0.5144V_{k} } \mathord{\left/ {\vphantom {{0.5144V_{k} } {(9.8065L)^{0.5} }}} \right. \kern-0pt} {(9.8065L)^{0.5} }}; \hfill \\ \, a = 4456.51 - 8105.61C_{B} + 4977.06C_{B}^{2} ; \hfill \\ \, b = - 6960.32 + 12817C_{B} - 10847.2C_{B}^{2} . \hfill \\ {\text{with 150}}{.0} \le L \le 274.32,{20}{\text{.0}} \le B \le 32.31, \hfill \\ { 13}{\text{.0}} \le D \le 25.0,10.0 \le T \le 11.71, \hfill \\ { 14}{\text{.0}} \le V_{k} \le 18.0,0.63 \le C_{B} \le 0.75. \hfill \\ \end{gathered}$$

1.8 Multi-product batch plant problem

$$\begin{gathered} {\text{Consider }}{\mathbf{x}} = [N_{1} ,N_{2} ,N_{3} ,V_{1} ,V_{2} ,V_{3} ,T_{L1} ,T_{L2} ,B_{1} ,B_{2} ] \hfill \\ {\text{Minimize }}f_{1} ({\mathbf{x}}) = \sum\limits_{j = 1}^{M} {\alpha_{j} N_{j} V_{j}^{{\beta_{j} }} } \hfill \\ \, f_{2} ({\mathbf{x}}) = 65\sum\limits_{i = 1}^{N} {\frac{{Q_{i} }}{{B_{i} }}} + 0.08Q_{1} + 0.1Q_{2} \hfill \\ \, f_{3} ({\mathbf{x}}) = \sum\limits_{i = 1}^{N} {\frac{{Q_{i} T_{Li} }}{{B_{i} }}} \hfill \\ {\text{subject to: }}g_{1} ({\mathbf{x}}) = f_{3} (\vec{x}) - H \le 0 \hfill \\ \, g_{2} ({\mathbf{x}}) = \sum\limits_{i = 1}^{N} {S_{ij} B_{i} - V_{j} } \le 0,j = 1,...,M \hfill \\ \, g_{3} ({\mathbf{x}}) = t_{ij} - N_{j} T_{Li} \le 0,i = 1,...,N;j = 1,...,M \hfill \\ {\text{where }}N = 2; \, M = 3; \, \alpha_{j} = 250; \, \beta_{j} = 0.6; \hfill \\ \, H = 6000; \, Q_{1} = 40000; \, Q_{2} = 20000; \hfill \\ \, S_{11} = 2; \, S_{12} = 3; \, S_{13} = 4; \hfill \\ \, S_{21} = 4; \, S_{22} = 6; \, S_{23} = 3; \hfill \\ \, t_{11} = 8; \, t_{12} = 20; \, t_{13} = 8; \hfill \\ \, t_{21} = 16; \, t_{22} = 4; \, t_{23} = 4. \hfill \\ {\text{with 1}} \le N_{1} ,N_{2} ,N_{3} \le 3({\text{integer}}),{250} \le V_{1} ,V_{2} ,V_{3} \le 2500, \hfill \\ { 6} \le T_{L1} \le 20,4 \le T_{L2} \le 16,{40} \le B_{1} \le 700,10 \le B_{2} \le 450. \hfill \\ \end{gathered}$$

Appendix 2. Pareto fronts obtained by all comparison algorithms

See Figs.

Fig. 20

Speed reducer design problem

20,

Fig. 21

Spring design problem

21,

Fig. 22

Vibrating platform design problem

22,

Fig. 23

Bulk carriers design problem

23,

Fig. 24

Multi-product batch plant design problem

24,

Fig. 25

72-bar 3D truss optimization problem

25,

Fig. 26

200-bar 2D truss optimization problem

26,

Fig. 27

942-bar 3D truss optimization problem

27